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# Integers: Factor = Divisor

If $$x$$ and $$y$$ are integers, is $$\frac{x}{y}$$ an integer? >(1) Both $$\frac{x}{4}$$ and $$\frac{x}{7}$$ are integers. >(2) $$\frac{y}{28}$$ is not an integer.
Correct. [[snippet]] According to Stat. (1), $$x$$ is divisible by 4 and 7, so $$x$$ is a multiple 28. However, that tells you nothing about $$y$$. Plug in values for $$x$$ and $$y$$: * If $$x = 28$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 56$$ and $$y = 3$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$\frac{y }{ 28}$$ is not an integer, so $$y$$ is not a multiple of 28. Alone, this tells you nothing about $$x$$ or $$y$$. Plug in values: * If $$x = 2$$ and $$y = 1$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 1$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(2) → Maybe → IS → CE**. According to Stat. (1+2), $$x$$ is a multiple of 28, and $$y$$ is not a multiple of 28. It seems as if $$x$$ is not divisible by $$y$$, so the answer is a definite "No," which is sufficient, but is that really the case? The variable $$y$$ can be any number as long as it is not divisible by 28. Plug in values: * If $$x = 28$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 28$$ and $$y = 3$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] According to Stat. (1), $$x$$ is divisible by 4 and 7, so $$x$$ is a multiple 28. However, that tells you nothing about $$y$$. Plug in values for $$x$$ and $$y$$: * If $$x = 28$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 56$$ and $$y = 3$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**. According to Stat. (2), $$\frac{y }{ 28}$$ is not an integer, so $$y$$ is not a multiple of 28. Alone, this tells you nothing about $$x$$ or $$y$$. Plug in values: * If $$x = 2$$ and $$y = 1$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 1$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(2) → Maybe → IS → CE**.
Incorrect. [[snippet]] According to Stat. (1+2), $$x$$ is a multiple of 28, and $$y$$ is not a multiple of 28. It seems as if $$x$$ is not divisible by $$y$$, so the answer is a definite "No," which is sufficient, but is that really the case? The variable $$y$$ can be any number as long as it is not divisible by 28. Plug in values: * If $$x = 28$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 28$$ and $$y = 3$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(1+2) → Maybe → IS → E**.
Incorrect. [[snippet]] According to Stat. (2), $$\frac{y }{ 28}$$ is not an integer, so $$y$$ is not a multiple of 28. Alone, this tells you nothing about $$x$$ or $$y$$. Plug in values: * If $$x = 2$$ and $$y = 1$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 1$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(2) → Maybe → IS → ACE**.
Incorrect. [[snippet]] According to Stat. (1), $$x$$ is divisible by 4 and 7, so $$x$$ is a multiple 28. However, that tells you nothing about $$y$$. Plug in values for $$x$$ and $$y$$: * If $$x = 28$$ and $$y = 2$$, then $$\frac{x}{ y}$$ is an integer. * If $$x = 56$$ and $$y = 3$$, then $$\frac{x}{ y}$$ is not an integer. There is no definite answer, so **Stat.(1) → Maybe → IS → BCE**.
Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.